We generalize to dimension p > 1 the notion of string structure and di
scuss the related obstruction. We apply our results to a model of boso
nic p-branes propagating on a principal G-bundle, coupled to a Yang-Mi
lls field and an antisymmetric tenser field and in the presence of a W
ess-Zumino term in the Lagrangian. We construct the quantization line
bundle and discuss the action of background gauge transformations on w
ave functions.